Finally, we address the (rare) transitions between different flow states for large \(\theta _m\) .
It was found that the Reynolds number ratio \(Re_z/Re_x\)  can well differentiate different convection roll states [1]}, [2]}. Here
\(Re_z = {\langle w^2\rangle _{V,t}^{1/2}}H/\nu \) 
is the volume averaged vertical Reynolds number and
\(Re_x = {\langle u^2\rangle _{V,t}^{1/2}}H/\nu \) 
the horizontal one. The Reynolds number ratios \(Re_z/Re_x\)  for \(Ra=10^{10}\)  with large \(\theta _m\)  close to \(\theta _m=1\)  are displayed in Fig. REF (a). The statistically stable two-roll state, as shown in Fig. REF (b), can be found for up to \(\theta _m=0.93\) . However, once \(\theta _m\)  increases to \(\theta _m=0.945\) , the two-roll state is not sustained all the time and rare transitions between two-roll and four-roll states occur, which can be seen in the rare jumps of the Reynolds number ratio \(Re_z/Re_x\)  in Fig. REF (b). The temperature fields for the two-roll and four-roll states are shown in Figs. REF (d) and REF (e), respectively. At even larger \(\theta _m=0.955\) , constant jumps between different states are observed, as displayed in Fig. REF (c). The instantaneous temperature fields at two instants are shown in Figs. REF (f) and REF (g). It can be seen that the convection rolls are now not well organised. When \(\theta _m\)  reaches 0.97, the flow enters into the conduction state without any fluid motions.
